I am so stuck on this question:
Consider a two-asset model where asset 0 is cash, so that the price of asset 0 is $B_t=1$ for all $t \geq0$. Asset 1 has prices given by $dS_t = a(S_t) dW_t$, where the given function $a$ is positive and smooth, and such $a$ and its derivative $a'$ is bounded. Let $\xi_t$ be the time-$t$ price of a European call option with maturity $T$ and strike $K$. Let $V: [0,T] \times \mathbb{R} \rightarrow \mathbb{R}_{+}$ satisfy the PDE (with boundary condition)
\begin{equation}
\frac{\partial V}{\partial t} (t,S) + \frac{a(S)^2}{2} \frac{\partial^2}{\partial S^2} V(t,S) =0, \quad V(T,S)= (S-K)^{+}.
\end{equation}
We let $\xi_t = V(t,S_t)$ so that there is no arbitrage.

3 Answers
3

I think the title here is misleading. Let's go back to the BS world with $r=0$ to $a(S_t)=S_t \sigma.$ In that case, all you are saying is that you can replicate a call option by holding $N(d_1)$ units of stock at time $t.$

What does this have to do with the second equation? I am guessing that this is the price process of an asset of nothing option with the stock taken as numeraire so it evaluates to $N(d_1).$

So my approach to this would be to repeat the BS replication argument when $\sigma$ is allowed to be a function of $S_t.$ Then work with the stock as numeraire to get the fact that the delta satisfies the second equation.

PS: I wrote $\delta^0_t$ and $\delta_t$ instead of $\phi_t,\pi_t$. The mistake in your self-financing equation is that it should write
$$
dV(t,S_t) = \pi_tdS_t + \phi_t dB_t
$$
but $B_t = 1$ so $dB_t = 0$ and we are left with
$$
\partial_S V(t,S_t) a(S_t)dW_t = U(t,S_t)a(S_t)dW_t
$$
which is another way of finding $\pi_t = U(t,S_t) = \partial_S V(t,S_t)$.

Without getting into all the Math one thing should be clear that:
Call option is equivalent to: long asset or nothing AND short cash or nothing options.

You cannot replicate a call option without asset or nothing since replicating portfolio for long call requires holding N(d1) quantity of the underlying. Asset or nothing gives you this exposure directly. Shorting cash or nothing which pays $K at maturity provides the remaining exposure required for replicating the long call option.